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Recent Advances in Theoretical and Numerical Analysis for Fractional and Integral Differential Equations

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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: 30 November 2024 | Viewed by 10611

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Special Issue Editors

Prof. Dr. Dongfang Li

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Guest Editor

Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, China
Interests: numerical solutions to PDE and ODE; fractional partial differential equation
Special Issues, Collections and Topics in MDPI journals

Dr. Hongyu Qin

E-Mail Website
Co-Guest Editor

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
Interests: convergence of numerical methods; diffusion; finite difference methods; iterative methods; numerical stability; partial differential equations
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Special Issue Information

Dear Colleagues,

Fractional differential equations and integral differential equations have attracted a great amount of attention in recent years. They widely appear in applied mathematics, physics, biology, chemistry and other disciplines. The typical models include sub-diffusion equations, diffusion-wave equations, space-fractional differential equations, and so on. It is usually difficult to obtain analytical solutions, due to the integral terms in the models. Fortunately, the evolution of differential equations can be well described by using some well-designed and high-order numerical schemes. Therefore, it has become a hot topic to numerically solve and analyze the equations.

In light of the aforementioned points regarding the significance of theoretical and numerical analysis, the potential topics include, but are not limited to, the following:

New theoretical results for fractional differential equations and integral differential equations;
New numerical methods for solving fractional differential equations;
New numerical methods for solving integral differential equations;
New numerical methods for solving non-local problems;
Numerical analysis of the numerical methods;
Application of fractional differential equations.

Prof. Dr. Dongfang Li
Dr. Hongyu Qin
Guest Editors

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Keywords

fractional differential equations
integral differential equations
theoretical analysis
numerical methods
numerical analysis

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Published Papers (11 papers)

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Research

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18 pages, 1608 KiB

Open AccessArticle

A Collocation Procedure for Treating the Time-Fractional FitzHugh–Nagumo Differential Equation Using Shifted Lucas Polynomials

by Waleed Mohamed Abd-Elhameed, Omar Mazen Alqubori and Ahmed Gamal Atta

Mathematics 2024, 12(23), 3672; https://doi.org/10.3390/math12233672 - 23 Nov 2024

Viewed by 301

Abstract

This work employs newly shifted Lucas polynomials to approximate solutions to the time-fractional Fitzhugh–Nagumo differential equation (TFFNDE) relevant to neuroscience. Novel essential formulae for the shifted Lucas polynomials are crucial for developing our suggested numerical approach. The analytic and inversion formulas are introduced, and after that, new formulas that express these polynomials’ integer and fractional derivatives are derived to facilitate the construction of integer and fractional operational matrices for the derivatives. Employing these operational matrices with the typical collocation method converts the TFFNDE into a system of algebraic equations that can be addressed with standard numerical solvers. The convergence analysis of the shifted Lucas expansion is carefully investigated. Certain inequalities involving the golden ratio are established in this context. The suggested numerical method is evaluated using several numerical examples to verify its applicability and efficiency. Full article

(This article belongs to the Special Issue Recent Advances in Theoretical and Numerical Analysis for Fractional and Integral Differential Equations)

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22 pages, 4183 KiB

Open AccessFeature PaperArticle

Exact Soliton Solutions to the Variable-Coefficient Korteweg–de Vries System with Cubic–Quintic Nonlinearity

by Hongcai Ma, Xinru Qi and Aiping Deng

Mathematics 2024, 12(22), 3628; https://doi.org/10.3390/math12223628 - 20 Nov 2024

Viewed by 359

Abstract

In this manuscript, we investigate the (2+1)-dimensional variable-coefficient Korteweg–de Vries (KdV) system with cubic–quintic nonlinearity. Based on different methods, we also obtain different solutions. Under the help of the wave ansatz method, we obtain the exact soliton solutions to the variable-coefficient KdV system, such as the dark and bright soliton solutions, Tangent function solution, Secant function solution, and Cosine function solution. In addition, we also obtain the interactions between dark and bright soliton solutions, between rogue and soliton solutions, and between lump and soliton solutions by using the bilinear method. For these solutions, we also give their three dimensional plots and density plots. This model is of great significance in fluid. It is worth mentioning that the research results of our paper is different from the existing research: we not only use different methods to study the solutions to the variable-coefficient KdV system, but also use different values of parameter t to study the changes in solutions. The results of this study will contribute to the understanding of nonlinear wave structures of the higher dimensional KdV systems. Full article

(This article belongs to the Special Issue Recent Advances in Theoretical and Numerical Analysis for Fractional and Integral Differential Equations)

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22 pages, 425 KiB

Open AccessFeature PaperArticle

Extension of the First-Order Recursive Filters Method to Non-Linear Second-Kind Volterra Integral Equations

by Rodolphe Heyd

Mathematics 2024, 12(22), 3612; https://doi.org/10.3390/math12223612 - 19 Nov 2024

Viewed by 429

Abstract

A new numerical method for solving Volterra non-linear convolution integral equations (NLCVIEs) of the second kind is presented in this work. This new approach, named IIRFM-A, is based on the combined use of the Laplace transformation, a first-order decomposition, a bilinear transformation, and the Adomian decomposition. Unlike most numerical methods based on the Laplace transformation, the IIRFM-A method has the dual advantage of requiring neither the calculation of the Laplace transform of the source function nor that of intermediate inverse Laplace transforms. The application of this new method to the case of non-convolutive multiplicative kernels is also introduced in this work. Several numerical examples are presented to illustrate the great flexibility and efficiency of this new approach. A concrete thermal problem, described by a non-linear convolutive Volterra integral equation, is also solved numerically using the new IIRFM-A method. In addition, this new approach extends for the first time the field of use of first-order recursive filters, usually restricted to the case of linear ordinary differential equations (ODEs) with constant coefficients, to the case of non-linear ODEs with variable coefficients. This extension represents a major step forward in the field of recursive filters. Full article

(This article belongs to the Special Issue Recent Advances in Theoretical and Numerical Analysis for Fractional and Integral Differential Equations)

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12 pages, 903 KiB

Open AccessArticle

A Novel and Accurate Algorithm for Solving Fractional Diffusion-Wave Equations

by Haifa Bin Jebreen and Ioannis Dassios

Mathematics 2024, 12(21), 3307; https://doi.org/10.3390/math12213307 - 22 Oct 2024

Viewed by 577

Abstract

The main objective of this work is to apply a novel and accurate algorithm for solving the second-order and fourth-order fractional diffusion-wave equations (FDWEs). First, the desired equation is reduced to the corresponding Volterra integral equation (VIE). Then, the collocation method is applied, for which the Chebyshev cardinal functions (CCFs) have been considered as the bases. In this paper, the CCFs based on a Lobatto grid are introduced and used for the first time to solve these kinds of equations. To this end, the derivative and fractional integral operators are represented in CCFs. The main features of the method are simplicity, compliance with boundary conditions, and good accuracy. An exact analysis to show the convergence of the scheme is presented, and illustrative examples confirm our investigation. Full article

(This article belongs to the Special Issue Recent Advances in Theoretical and Numerical Analysis for Fractional and Integral Differential Equations)

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26 pages, 58312 KiB

Open AccessArticle

Comprehensive Numerical Analysis of Time-Fractional Reaction–Diffusion Models with Applications to Chemical and Biological Phenomena

by Kolade M. Owolabi, Sonal Jain, Edson Pindza and Eben Mare

Mathematics 2024, 12(20), 3251; https://doi.org/10.3390/math12203251 - 17 Oct 2024

Viewed by 748

Abstract

This paper aims to present a robust computational technique utilizing finite difference schemes for accurately solving time fractional reaction–diffusion models, which are prevalent in chemical and biological phenomena. The time-fractional derivative is treated in the Caputo sense, addressing both linear and nonlinear scenarios. The proposed schemes were rigorously evaluated for stability and convergence. Additionally, the effectiveness of the developed schemes was validated through various linear and nonlinear models, including the Allen–Cahn equation, the KPP–Fisher equation, and the Complex Ginzburg–Landau oscillatory problem. These models were tested in one-, two-, and three-dimensional spaces to investigate the diverse patterns and dynamics that emerge. Comprehensive numerical results were provided, showcasing different cases of the fractional order parameter, highlighting the schemes’ versatility and reliability in capturing complex behaviors in fractional reaction–diffusion dynamics. Full article

(This article belongs to the Special Issue Recent Advances in Theoretical and Numerical Analysis for Fractional and Integral Differential Equations)

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35 pages, 702 KiB

Open AccessFeature PaperArticle

Numerical Solution of Linear Second-Kind Convolution Volterra Integral Equations Using the First-Order Recursive Filters Method

by Rodolphe Heyd

Mathematics 2024, 12(15), 2416; https://doi.org/10.3390/math12152416 - 3 Aug 2024

Viewed by 644

Abstract

A new numerical method for solving Volterra linear convolution integral equations (CVIEs) of the second kind is presented in this work. This new approach uses first-order infinite impulse response digital filters method (IIRFM). Three convolutive kernels were analyzed, the unit kernel and two singular kernels: the logarithmic and generalized Abel kernels. The IIRFM is based on the combined use of the Laplace transformation, a first-order decomposition, and a bilinear transformation. This approach often leads to simple analytical expressions of the approximate solutions, enabling efficient numerical calculation, even using single-precision floating-point numbers. When compared with the method of homotopic perturbations with Laplace transformation (HPM-L), the IIRFM approach does not present, in linear cases, the convergence difficulties inherent to iterative approaches. Unlike most solution methods based on the Laplace transform, the IIRFM has the dual advantage of not requiring the calculation of the Laplace transform of the source function, and of not requiring the systematic calculation of inverse Laplace transforms. Full article

(This article belongs to the Special Issue Recent Advances in Theoretical and Numerical Analysis for Fractional and Integral Differential Equations)

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23 pages, 349 KiB

Open AccessArticle

Inequalities for Riemann–Liouville-Type Fractional Derivatives of Convex Lyapunov Functions and Applications to Stability Theory

by Ravi P. Agarwal, Snezhana Hristova and Donal O’Regan

Mathematics 2023, 11(18), 3859; https://doi.org/10.3390/math11183859 - 9 Sep 2023

Cited by 1 | Viewed by 930

Abstract

In recent years, various qualitative investigations of the properties of differential equations with different types of generalizations of Riemann–Liouville fractional derivatives were studied and stability properties were investigated, usually using Lyapunov functions. In the application of Lyapunov functions, we need appropriate inequalities for the fractional derivatives of these functions. In this paper, we consider several Riemann–Liouville types of fractional derivatives and prove inequalities for derivatives of convex Lyapunov functions. In particular, we consider the classical Riemann–Liouville fractional derivative, the Riemann–Liouville fractional derivative with respect to a function, the tempered Riemann–Liouville fractional derivative, and the tempered Riemann–Liouville fractional derivative with respect to a function. We discuss their relations and their basic properties, as well as the connection between them. We prove inequalities for Lyapunov functions from a special class, and this special class of functions is similar to the class of convex functions of many variables. Note that, in the literature, the most common Lyapunov functions are the quadratic ones and the absolute value ones, which are included in the studied class. As a result, special cases of our inequalities include Lyapunov functions given by absolute values, quadratic ones, and exponential ones with the above given four types of fractional derivatives. These results are useful in studying types of stability of the solutions of differential equations with the above-mentioned types of fractional derivatives. To illustrate the application of our inequalities, we define Mittag–Leffler stability in time on an interval excluding the initial time point. Several stability criteria are obtained. Full article

(This article belongs to the Special Issue Recent Advances in Theoretical and Numerical Analysis for Fractional and Integral Differential Equations)

21 pages, 9043 KiB

Open AccessEditor’s ChoiceArticle

An Efficient Numerical Approach for Solving Systems of Fractional Problems and Their Applications in Science

by Sondos M. Syam, Z. Siri, Sami H. Altoum and R. Md. Kasmani

Mathematics 2023, 11(14), 3132; https://doi.org/10.3390/math11143132 - 16 Jul 2023

Cited by 10 | Viewed by 1450

Abstract

In this article, we present a new numerical approach for solving a class of systems of fractional initial value problems based on the operational matrix method. We derive the method and provide a convergence analysis. To reduce computational cost, we transform the algebraic [...] Read more.

2 \times 2

nonlinear equations, instead of solving a system of 2 m × 2 m equations. We apply our approach to three main applications in science: optimal control problems, Riccati equations, and clock reactions. We compare our results with those of other researchers, considering computational time, cost, and absolute errors. Additionally, we validate our numerical method by comparing our results with the integer model when the fractional order approaches one. We present numerous figures and tables to illustrate our findings. The results demonstrate the effectiveness of the proposed approach. Full article

(This article belongs to the Special Issue Recent Advances in Theoretical and Numerical Analysis for Fractional and Integral Differential Equations)

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10 pages, 274 KiB

Open AccessArticle

A Second-Order Time Discretization for Second Kind Volterra Integral Equations with Non-Smooth Solutions

by Boya Zhou and Xiujun Cheng

Mathematics 2023, 11(12), 2594; https://doi.org/10.3390/math11122594 - 6 Jun 2023

Viewed by 977

Abstract

In this paper, a novel second-order method based on a change of variable and the symmetrical and repeated quadrature formula is presented for numerical solving second kind Volterra integral equations with non-smooth solutions. Applying the discrete Grönwall inequality with weak singularity, the convergence [...] Read more.

O (N^{- 2})

L^{\infty}

norm is proved, where N refers to the number of time steps. Numerical results are conducted to verify the efficiency and accuracy of the method. Full article

(This article belongs to the Special Issue Recent Advances in Theoretical and Numerical Analysis for Fractional and Integral Differential Equations)

17 pages, 4635 KiB

Open AccessArticle

Mixed Convection Flow of Water Conveying Graphene Oxide Nanoparticles over a Vertical Plate Experiencing the Impacts of Thermal Radiation

by Umair Khan, Aurang Zaib, Anuar Ishak, Iskandar Waini and Ioan Pop

Mathematics 2022, 10(16), 2833; https://doi.org/10.3390/math10162833 - 9 Aug 2022

Cited by 2 | Viewed by 1579

Abstract

Water has drawn a lot of interest as a manufacturing lubricant since it is affordable, eco-friendly, and effective. Due to their exceptional mechanical qualities, water solubility, and variety of application scenarios, graphene oxide (GO)-based materials have the potential to increase the lubricant performance of water. The idea of this research was to quantify the linear 3D radiative stagnation-point flow induced by nanofluid through a vertical plate with a buoyancy or a mixed convection effect. The opposing, as well as the assisting, flows were considered in the model. The leading partial differential equations (PDEs) were transformed into dimensionless similarity equations, which were then solved numerically via a bvp4c solver. The influences of various physical constraints on the fluid flow and thermal properties of the nanofluid were investigated and are discussed. Water-based graphene oxide nanoparticles were considered in this study. The numerical outcomes indicated that multiple solutions were obtained in the case of the opposing flow (λ < 0). The critical values increased as the nanoparticle volume fraction became stronger. Furthermore, as the nanoparticles increased in strength, the friction factor increased and the heat transfer quickened. The radiation factor escalated the heat transfer in both solutions. In addition, a temporal stability analysis was also undertaken to verify the results, and it was observed that the branch of the first outcome became physically reliable (stable) whilst the branch of the second outcome became unstable, as time passed. Full article

(This article belongs to the Special Issue Recent Advances in Theoretical and Numerical Analysis for Fractional and Integral Differential Equations)

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Review

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12 pages, 439 KiB

Open AccessReview

A Mini-Review on Recent Fractional Models for Agri-Food Problems

by Stefania Tomasiello and Jorge E. Macías-Díaz

Mathematics 2023, 11(10), 2316; https://doi.org/10.3390/math11102316 - 16 May 2023

Cited by 1 | Viewed by 1322

Abstract

This work aims at providing a concise review of various agri-food models that employ fractional differential operators. In this context, various mathematical models based on fractional differential equations have been used to describe a wide range of problems in agri-food. As a result of this review, we found out that this new area of research is finding increased acceptance in recent years and that some reports have employed fractional operators successfully in order to model real-world data. Our results also show that the most commonly used differential operators in these problems are the Caputo, the Caputo–Fabrizio, the Atangana–Baleanu, and the Riemann–Liouville derivatives. Most of the authors in this field are predominantly from China and India. Full article

(This article belongs to the Special Issue Recent Advances in Theoretical and Numerical Analysis for Fractional and Integral Differential Equations)

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